Abstract
We continue the study of the performance of mildly greedy players in cut games initiated by Christodoulou et al. (Theoret Comput Sci 438:13–27, 2012), where a mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than \(1+\epsilon \), for some given \(\epsilon \ge 0\). Hence, in presence of mildly greedy players, the classical concepts of pure Nash equilibria and best-responses generalize to those of \((1+\epsilon )\)-approximate pure Nash equilibria and \((1+\epsilon )\)-approximate best-responses, respectively. We first show that the \(\epsilon \)-approximate price of anarchy, that is the price of anarchy of \((1+\epsilon )\)-approximate pure Nash equilibria, is at least \(\frac{1}{2+\epsilon }\) and that this bound is tight for any \(\epsilon \ge 0\). Then, we evaluate the approximation ratio of the solutions achieved after a \((1+\epsilon )\)-approximate one-round walk starting from any initial strategy profile, where a \((1+\epsilon )\)-approximate one-round walk is a sequence of \((1+\epsilon )\)-approximate best-responses, one for each player. We improve the currently known lower bound on this ratio from \(\min \left\{ \frac{1}{4+2\epsilon },\frac{\epsilon }{4+2\epsilon }\right\} \) up to \(\min \left\{ \frac{1}{2+\epsilon },\frac{2\epsilon }{(1+\epsilon )(2+\epsilon )}\right\} \) and show that this is again tight for any \(\epsilon \ge 0\). An interesting and quite surprising consequence of our results is that the worst-case performance guarantee of the very simple solutions generated after a \((1+\epsilon )\)-approximate one-round walk is the same as that of \((1+\epsilon )\)-approximate pure Nash equilibria when \(\epsilon \ge 1\) and of that of subgame perfect equilibria (i.e., Nash equilibria for greedy players with farsighted, rather than myopic, rationality) when \(\epsilon =1\).
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This work was partially supported by the PRIN 2010–2011 research project ARS TechnoMedia: “Algorithmics for Social Technological Networks” funded by the Italian Ministry of University
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Bilò, V., Paladini, M. On the performance of mildly greedy players in cut games. J Comb Optim 32, 1036–1051 (2016). https://doi.org/10.1007/s10878-015-9898-2
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DOI: https://doi.org/10.1007/s10878-015-9898-2